No Crowning at the Frontier: The Halting Structure of Finite Explanation

Any actually delivered explanation must halt. But halting is not the same as certifying the point at which explanation halts as a crownable terminus. This paper argues that, for any explainer, such internal crowning cannot be completed. Taking the delivered explanatory graph as the unit of analysis, it establishes three claims. First, every successfully delivered explanatory graph must contain stopping points, formalized below as frontier nodes. Second, frontier status and completed certification are structurally incompatible: once a stopping point is certified, it immediately ceases to be a stopping point, and the certifying basis becomes the new stopping point in the graph. Third, the inferential principles that license explanatory steps within the graph cannot themselves be certified within the same explanatory framework, because any such attempt either appeals to a higher-order principle or presupposes the very principle it seeks to authorize. The paper distinguishes certification from defence: non-frontier nodes can typically be certified, whereas stopping points cannot be certified and can only be defended. Explanations often succeed, but no explanatory graph can internally complete the final crowning of its own stopping points or inferential principles.